1fa1etutorial 2

7/30/2019 1fa1eTutorial 2

1/18

AMITY SCHOOL OF ENGINEERING & TECHNOLOGY

Control Systems (BTEEC-30505)

1

Tutorial-3

1. Obtain the transfer function of the mechanical system shown below

a)

b)

2. Draw the mechanical network of the mechanical system and write the nodal equations.

3. Construct the mechanical network for the mechanical vibration absorber and write the equations

of performance


2/18



2

4. Obtain the transfer function of the mechanical system shown

a) b)

5. Write the differential equations and transfer function I(s)/Vi(s) for the electrical circuita)

b)

c)


3/18



3

6. Obtain transfer function Vo(s)/Vi(s) for the electrical circuit and plot its poles and zeros for R=C=1

7. Write the differential equations and transfer function Vo(s)/Vi(s) forthe electrical circuit. Assume

T=RC

8. Write the differential equations and transfer function Vo(s)/Vi(s) forthe electrical circuit. Assume

L/R=T1 and CR=T2


4/18



4

9. Obtain transfer function for the electrical circuit in the form.

If Vi= 8sin10t, R1=50 K, R2= 5K, and C=1F, calculate output voltage in magnitude and phase,

relative to input voltage. For what values of R1 and R2 will give T=0.6 sec and a=0.1

10. Obtain the nodal equations of the mechanical system. Draw its mechanical network and find the

transfer function. Also, draw analogous circuit based on force currentanalogy

11. Draw mechanical network of the mechanical systems, write the nodal equations and draw the

analogous circuit based on force voltage analogy.

1

21

2 andwhere1

)1(

)(

)(CRT

RR

Ra

asT

sTa

sV

sV

i

o


5/18



5

12. Obtain the nodal equations of the mechanical system. Draw its mechanical

network and analogous circuit based on force voltage analogy

13. Obtain the nodal equations of the mechanical system. Draw its mechanical

network and analogous circuit based on force voltage analogy

14. Obtain the transfer function of the mechanical system and draw the electrical analogous

network.

15. Find the transfer function of a control system described by

Where r(t) and c(t) are input and output respectively.

)2()()(

2 trtcdt

tdc


6/18



6

Tutorial-4

1. Find the time response, initial values and final values of the following

functions

)3()2(

)1(12)(

)6)(4)(2(

)10(

)(

2

sss

ssF

sss

ss

sF

2. Find the unit step response of a control system as given below.Also find its transfer function.

tettc

2

2

55

2

5)(

3. Give an example of Type 0, Type 1 and Type 2 Systems

4. Prove that the output acceleration is proportional to the error angle

1

2

2

( ) (1 )( )

( ) (1 )

accelaration constant

o a

c

a

s K sT G s

s s sT

where

K

5. A servo system for the position control of a rotatable mass is stabilized by viscous damping which

is three-quarters of that is needed for critical damping. The undamped natural frequency of

oscillations is 12 Hz. Derive an expression for the input of the system, if the input control issuddenly moved to a new position, being initially at rest. Hence, find the maximum overshoot.

6. A servomechanism is represented by the following equation where E=(r-) is the actuating signal.

Calculate the value of damping ratio, undamped and damped frequency of oscillations.

2

210 150

d dE

dt dt

7. Measurements conducted on a servomechanism show the following system response when

subjected to a unit step input. Obtain the expression for the closed-loop transfer function,

damping ratio and undamped frequency of oscillations.

60 10( ) 1 0.2 1.2

t tc t e e

1.A unity feedback system has an open-loop transfer function as G(s) =K/s(s+10). Determine thevalue of K so that the damping ratio is 0.5. For this vale of K, find settling time, peak overshoot

and time to peak overshoot for a unit step input.


7/18



7

9. The open loop transfer function of unity feedback system is given by G(s) =K/s (1+sT) where T and

K are constants having positive values. By what factor the amplifier gain is reduced so that (a)

the peak overshoot for a unit step response is reduced from 75 to 25 percent (b) the damping

ratio increases from 0.1 to 0.6.

10. The control system shown in fig below employs proportional plus error rate control. Determine

the value of error rate constant Ke so that the damping ratio is 0.6.Determine the values of

settling time, maximum overshoot and steady state error, if the input is unit ramp. What will be

the value of steady state error without error rate control?

Tutorial-5


8/18



8

1. A second order position control system has the following open loop transfer function where A is

the amplifier gain. Find the value of A so that steady state error does not exceed one degree

when the input shaft rotates at 10 rpm.

G(s) =5.73A/s (1+0.1s),

2. Considering step input with an error angle of e and K as the gain constant, find the steady state

response of a control system whose open loop transfer function is given as

0

2

( 2)( )

( 4 8)e

K sG s

s s s

3. Determine damping ratio, natural undamped frequency, peak overshoot and

expression for error response for a unity feedback system whose closed loop transfer function is

2

( ) 5

( ) 4 5

C s

R s s s

4. Find all time domain specifications for a unity feedback control system whose open loop transfer

function is given as

25

( )( 6)

G ss s

5. Determine the error coefficients and steady state error for a unity and non-unity feedback

systems

( ) 1( ) ( 2)

( ) ( 1)( 10)

C sH s s

R s s s s

6.Show that if the input is step displacement, the output will complete 98.26% of the step in 6T secsfor critical damping for a system whose transfer function is

2

1

(1 )sT

7. A servo system for position control has the closed-loop transfer function given below. Find the

percentage peak overshoot; if the input is suddenly moved to a new position.

2

( ) 6

( ) 2 6

C s

R s s s

8. The forward transfer function of a unity feedback type 1 second order system has a pole at -2. The

nature of gain K is so adjusted that the damping ratio is 0.4.The system is subjected to input r (t)

= 1+4t. Find the steady state error.


9/18



9

Tutorial-6

1.Sketch the polar plot for the following( )

1

sTG s

sT


10/18



10

2.Comment on the stability by applying Nyquist stability criterion

2

4 1( ) ( ) ( )

( 1)(2 1)

1( ) ( ) ( )(1 2 )(1 )

2( ) ( ) ( )

( 1)( 1)

( 3)( ) ( ) ( )

( 1)

1( ) ( ) ( )

2 (1 20 )

5( ) ( ) ( )

(1 )

sa G s H s

s s s

b G s H ss s s

sc G s H s

s s

K sd G s H s

s s

e G s H ss s

f G s H ss s

3.The closed loop transfer function of a feedback system is given by

2

( ) 1000

( ) ( 22.5)( 2.45 44.4)

C s

R s s s s

a. Determine the resonance peak and resonant frequency of the system by drawing thefrequency response curve.

b. What should be the values of damping ratio and undamped natural frequency ofoscillations of an equivalent second order system which will produce same resonance

peak and resonant frequency of oscillations?

c. Determine the bandwidth of the equivalent second order system4.Sketch the root locus for the following transfer functions

2

2

2

2

( 1)( ) ( )

( 3)( 5)

( 1)

( ) ( )

6( ) ( )

2 6

40( ) ( )

( 20)( 60 10000)

K sa G s

s s s

K s

b G s s

c G ss s

sd G s

s s s s

5. Plot the root-locus pattern of a system whose forward path transfer function is


11/18



11

G(s) =

6. Plot the root-locus pattern of a system whose forward path transfer function is

G(s) =

7. A unity feed-back control system has

G(s) =

Sketch the root locus and show on it

(a) Breakaway point

(b) Line for = 0.5 and value ofKfor this damping ratio (c) The frequency at which the root

locus crosses the imaginary axis and the corresponding value ofK.

8. The loop transfer function of a feedback control system is given by

G(s)H(s) =

(a) Sketch the root locus plot with Kas a variable parameter and show that loci of complex roots

are part of a circle.

(b) Determine the breakaway/break-in points, if any.

(c) Determine the range ofKfor which the system is underdamped.

(d) Determine the value ofKfor critical damping

(e) Determine the minimum value of damping ratio.

9. For a control system in figure below, draw the root locus diagram neatly on a graph paper

showing all the relevant calculations. From the diagram comment on how the value of K will

make the system under-damped or over-damped


12/18



12

10. Consider the system shown in figure below. Draw the locus of the poles of the overall system

as Kis varied from zero to infinity.

11. Sketch the root locus plot for a control system represented by

G(s)H(s) =

Tutorial-7

Q1) The open loop transfer function of a feedback control system is given by

G(s)H(s)=

The parameters K and T may be represented in a plane with K as the horizontal axis and T as vertical

axis .Determine the region in which closed loop system is stable.

Q2) Using Rouths stability criterion, ascertain stability for each of the following cases.

a)

b)


13/18



13

c)6 5 4 3 23 4 6 5 3 2 0s s s s s s

Q3) The characteristic equation of a feedback control system is3 2

3 ( 2) 4 0s ks k s .

Determine the range of k for which system is stable.

Q4) The characteristic equation of a feedback control system is4 3 2

20 5 10 15 0s ks s s Find the

range of k for which system is stable.

Q5) Find the conditions for stability for the systems whose characteristic equations are given below.

The case where stability is suggested for real values of K, determine the values of K which will cause

sustained oscillations .find the frequency of oscillations

a)4 3 2

20 224 1240 2400 0s s s s k

b)3 2

( 0.5) 4 50 0s k s ks

Q6) The open loop transfer function of a unity feedback control system is given as

( ) ( )(1 )

KG s H s

s sT

It is desired that all roots of the characteristics equation must lie in the region to the left of the line

s=-a. Determine the values of K and T required so that there are no roots to right of the line s=-a.

Q7) The open loop transfer function of a unity feedback system is given by

2( )

( 3)( 1)KG s

s s s s

Determine the values of K that will cause sustained oscillations in the closed loop system. Also find

the oscillation frequency.

Q8) A control system is depicted in fig below. If K=2 find out how many times the gain may be increased

before on stability access.

Q9) Determine the values of K and B so that the system whose open loop transfer- function is


14/18



14

3 2

( 1)( )

3 1

K sG s

s bs s

Oscillates at a frequency of 2 rad./sec. Assume unity Feedback.

Q10) A position control system is shown in the fig below.

K and a are the parameters of the system. Determine the range of K and a for which the system

is stable.

Tutorial-8

Q1) Sketch the Bode Plot and determine the gain Cross-over Frequency and Phase Cross over

frequencies.

10( )

( 0.5 )(1 0.1 )G s

s s s s

Q2) Sketch the Bode Plot for the transfer Function

2

( )(1 0.2 )(1 0.02 )

KsG s

s s

Q3) Sketch the Bode Plot for a unity feedback system characterized by the open loop transfer function


15/18



15

2

(1 0.2 )(1 0.025 )( )

(1 0.001 )(1 0.005 )

K s sG s

s s s

Show that the system is continually stable. Find the range of values of K for which the system is stable.

Q4) Draw the Bode Plot for a system having

100( )

(1 )(2 )G s

s s s

Find a) Gain Margin

b) Phase Margin

c) Gain Cross over Frequency

d) Phase Crossover Frequency

Q5) The open loop transfer function of a unity feedback is

( )(2 )(10 )

KG s

s s s

Construct Bode Plot and determine

a) Limiting Value of K for system to be stable

b) Value of Gain margin to be 10dB

c) Value of Phase margin to be 50 .

Q6) The open Loop Transfer Function of a unity Feedback system is

1

( )(0.5 1)(0.1 1)

G ss s s

Find Gain and Phase Margin. if Phase lag element with transfer function of (1+2s)/(1+5s) is added in

the forward path find how much the gain must be changed to keep the margin same.

Q7) Consider the system shown in Fig below. Design Lead Compensator of this system to meet the

following specification

Damping Ratio =0.7

Settling Time =1.4 sec

Velocity Error Constant =2


16/18



16

Q8) Find the open loop transfer function of a system whose approximate plot is shown as below

Q9) Determine the Transfer Function whose approximate plot is shown below.

Q10) Derive the Transfer Function of the system from the data given on the Bode Diagram shown in fig

below.


17/18



17

Tutorial-9

Q1) Explain need for compensator and define various types of compensators.

Q2) Obtain transfer function of phase lead compensator and derive max phase lead angle with bode

plot.

Q3) Derive parameter of Phase lag compensator and show properties.

Q4) What is phase lag lead compensation? Show various properties of this type of compensator with

Bode plot.

Q5) Show different modes of feedback compensator

Q6) Design a suitable lag compensating network for G(s) = to meet the following

specifications. Kv (velocity constant)= 20sec-1

,phase margin35.

Q7) The open loop transfer function of a type 2 system with unity feedback is given by

G(s) =

Design a lead compensator to meet the following specifications KA (acceleration constant) =10sec-

2, phase margin=35.


18/18



18

Q8) Design a suitable lag compensating network for G(s) = to meet the following

specifications. Kv (velocity constant) 5sec-1damping ratio= 0.5, settling time ts =10 sec.

1fa1etutorial 2

Documents